<ul><li>Motivated by robust and quantile regression problems, a study investigates the stochastic gradient descent (SGD) algorithm for minimizing an objective function with a sub--quadratic tail.</li><li>The study introduces a novel piecewise Lyapunov function that can handle functions with only first-order differentiability, including popular loss functions such as Huber loss.</li><li>Finite-time moment bounds are derived for general diminishing stepsizes and constant stepsizes, and weak convergence, central limit theorem, and bias characterization are established for constant stepsize.</li><li>The results have wide applications, specifically in online robust regression and online quantile regression.</li></ul>

A Piecewise Lyapunov Analysis of sub--quadratic SGD: Applications to Robust and Quantile Regression

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